Integral of \( \displaystyle \sin^{3}{\left(2 x \right)} \)

Problem 4.647 · easy

1. \[ \int \sin^{3}{\left(2 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(1 - \cos^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}\, dx \]
   trig-identityUse the identity sin(2x)^2 = 1 - cos(2x)^2.✓ Proved
3. \[ = - \int \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)}\, dx + \int \sin{\left(2 x \right)}\, dx \]
   linearityDistribute the sin(2x) term.✓ Proved
4. \[ = - \frac{\cos{\left(2 x \right)}}{2} - \int \sin{\left(2 x \right)} \cos^{2}{\left(2 x \right)}\, dx \]
   antiderivativeIntegrate the first term.✓ Proved
5. \[ = \frac{\cos^{3}{\left(2 x \right)}}{6} - \frac{\cos{\left(2 x \right)}}{2} \]
   antiderivative simplifyIntegrate the second term using substitution logic. Simplify the signs and fractions.✓ Proved